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| Mirrors > Home > MPE Home > Th. List > imass1 | Structured version Visualization version GIF version | ||
| Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
| Ref | Expression |
|---|---|
| imass1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssres 6021 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | |
| 2 | rnss 5950 | . . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) |
| 4 | df-ima 5698 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
| 5 | df-ima 5698 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
| 6 | 3, 4, 5 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: predrelss 6358 vdwnnlem1 17033 dprdres 20048 imasnopn 23698 imasncld 23699 imasncls 23700 utoptop 24243 restutop 24246 ustuqtop3 24252 utopreg 24261 metustbl 24579 imadifxp 32614 gsumfs2d 33058 esum2d 34094 eulerpartlemmf 34377 bj-imdirco 37191 brtrclfv2 43740 frege97d 43765 frege109d 43770 frege131d 43777 hess 43793 resimass 45246 setrecsss 49220 |
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